Coursework - Data Science II

Introduction

This coursework focuses on housing prices, with the main objective being to predict the price of a property based on various inputs. The inputs include features such as the area, the number and types of rooms, and additional factors like the availability of a main road, hot water heating, and more.

The dependent variable is the price, as it is the primary concern for most people searching for a house. The goal of this work is to predict the price based on diverse inputs, which consist of mixed data types, such as:

• Numerical values
• Text-based responses like “yes” or “no”
• Categories for furnishing status, including “furnished,” “semi-furnished,” or “non-furnished.”

This project addresses a regression problem because the objective is to predict a numeric value—in this case, the price of the property.

Description

• Collect your dataset(s), explore your data for deficiencies such as missing data and formatting problems and prepare it for modelling.
• Extensive data collection and preparation yields extra credit but is not mandatory for this coursework.
• Explore the data via descriptive statistics and visualization.

Collection

Here I would like to collect, prepare, and explore my data. First thing is to import the data set.

dt_houses <- fread(file = "./data/Regression_set.csv")

I would like to check, if i have some nullish data in my dataset. I think it is a good idea to go through all rows and colums and check, if there is a NA. I want to check it with built-in function in R complete.cases(data_table). This function returns TRUE or FALSE if row contains a NA value.

nas <- dt_houses[!complete.cases(dt_houses)]
nas

That looks great. Now we can move on to exploration. But before I start, It is crucial to install all needed libraries.

library(data.table)
library(ggcorrplot)
library(ggExtra)
library(ggplot2)
library(ggridges)
library(ggsci)
library(ggthemes)
library(RColorBrewer)
library(svglite)
library(viridis)
library(scales)
library(rpart)
library(rpart.plot)

Exploration

I found some helpful functions in R, so we could have a look on our data. We will start with a structure, than we will get some statistic data and take a head() of the data

str(dt_houses)

Classes ‘data.table’ and 'data.frame':  545 obs. of  13 variables:
 $ price           : int  13300000 12250000 12250000 12215000 11410000 10850000 10150000 10150000 9870000 9800000 ...
 $ area            : int  7420 8960 9960 7500 7420 7500 8580 16200 8100 5750 ...
 $ bedrooms        : int  4 4 3 4 4 3 4 5 4 3 ...
 $ bathrooms       : int  2 4 2 2 1 3 3 3 1 2 ...
 $ stories         : int  3 4 2 2 2 1 4 2 2 4 ...
 $ mainroad        : chr  "yes" "yes" "yes" "yes" ...
 $ guestroom       : chr  "no" "no" "no" "no" ...
 $ basement        : chr  "no" "no" "yes" "yes" ...
 $ hotwaterheating : chr  "no" "no" "no" "no" ...
 $ airconditioning : chr  "yes" "yes" "no" "yes" ...
 $ parking         : int  2 3 2 3 2 2 2 0 2 1 ...
 $ prefarea        : chr  "yes" "no" "yes" "yes" ...
 $ furnishingstatus: chr  "furnished" "furnished" "semi-furnished" "furnished" ...
 - attr(*, ".internal.selfref")=<externalptr> 

Statistic data:

summary(dt_houses[, .(price, area, bedrooms, bathrooms, stories, parking)])

     price               area          bedrooms       bathrooms        stories         parking      
 Min.   : 1750000   Min.   : 1650   Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :0.0000  
 1st Qu.: 3430000   1st Qu.: 3600   1st Qu.:2.000   1st Qu.:1.000   1st Qu.:1.000   1st Qu.:0.0000  
 Median : 4340000   Median : 4600   Median :3.000   Median :1.000   Median :2.000   Median :0.0000  
 Mean   : 4766729   Mean   : 5151   Mean   :2.965   Mean   :1.286   Mean   :1.806   Mean   :0.6936  
 3rd Qu.: 5740000   3rd Qu.: 6360   3rd Qu.:3.000   3rd Qu.:2.000   3rd Qu.:2.000   3rd Qu.:1.0000  
 Max.   :13300000   Max.   :16200   Max.   :6.000   Max.   :4.000   Max.   :4.000   Max.   :3.0000  

and this is a sample of dataset:

head(dt_houses)

I would like to start from density of a main values, which are from my domain knowledge are important in price of the properties

Price density:

ggplot(data = dt_houses, aes(x = price)) + 
  geom_density(fill="#f1b147", color="#f1b147", alpha=0.25) + 
  labs(
    x = 'Price',
    y = 'Density'
  ) +
  geom_vline(xintercept = mean(dt_houses$price), linetype="dashed") + 
  scale_x_continuous(labels = label_number(scale = 1e-6, suffix = "M")) + 
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

This density plot visualizes the distribution of house prices, showing that most houses are priced around 4-5 million, with a right-skewed distribution (some higher-priced houses pulling the mean up). The dashed vertical line represents the mean price (~5M). The plot highlights that while most houses fall within a moderate price range, some expensive properties extend beyond 10M.

Area density:

ggplot(data = dt_houses, aes(x = area)) + 
  geom_density(fill="#f1b147", color="#f1b147", alpha=0.25) + 
  labs(
    x = 'Price',
    y = 'Density'
  ) +
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

The area density plot looks similar to price density plot and can also make sense, because if house has a bigger area, the higher cost is quite expected. This plot shows that most houses are having area in range ~3000-5000. But some properties have area more than 12000.

Next plot will visualize the distribution of price depending on area.

ggplot() + 
  geom_point(data = dt_houses, aes(x = area, y = price, color = parking)) +
  scale_y_continuous(labels = label_number(scale = 1e-6, suffix = "M")) + 
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

This scatter plot visualizes the relationship between house area (x-axis) and price (y-axis), with color indicating the number of parking spaces. It shows a positive correlation between area and price—larger houses tend to be more expensive. However, there is some variability, as some large houses have relatively lower prices. The color gradient suggests that houses with more parking spaces (lighter blue) tend to be higher in price and larger in area.

The next plot, which I am going to do is a boxplot and I want to use bedrooms as a factor variable on x axis and price on y-axis, to get an overall understanding, how amount of bedrooms affect price.

ggplot(data = dt_houses, aes(x = factor(bedrooms), y = price)) +
  geom_boxplot() + 
  theme_minimal() 

Boxplot shows, that on average, houses with more bedrooms have higher prices, but around 4-6 bedrooms, 1 quantile stagnates, and so does median price. There are some outliers, but not too much.

It is also interesting to take a look at distribution of bedrooms, so next plot would be a histogram, because I want to know, which amount of bedrooms is the most “popular” in the whole dataset.

ggplot(data = dt_houses, aes(x = bedrooms)) + 
  geom_histogram(fill="#2f9e44", color="#2f9e44", alpha=0.25) + 
  geom_vline(xintercept = mean(dt_houses$bedrooms), linetype="dashed") + 
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

mean of the bedrooms:

mean(dt_houses$bedrooms)

[1] 2.965138

From this visualization we can mention, that the most of the houses have 2, 3 or 4 rooms. 1, 5 and 6 rooms are not as popular in this dataset.

Let’s have a look at histogram of stories:

ggplot(data = dt_houses, aes(x = stories)) + 
  geom_histogram(fill="#2f9e44", color="#2f9e44", alpha=0.25) + 
  geom_vline(xintercept = mean(dt_houses$stories), linetype="dashed") + 
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

mean(dt_houses$stories)

[1] 1.805505

This plot shows that most popular amount of stories are 1 and 2. 3 and 4 makeing less than 100 houses together.

Bathrooms are also interesting variable, so let’s take a look at histogram and a Boxplot bathrooms and price:

ggplot(data = dt_houses, aes(x = bathrooms)) + 
  geom_histogram(fill="#2f9e44", color="#2f9e44", alpha=0.25) + 
  geom_vline(xintercept = mean(dt_houses$bathrooms), linetype="dashed") + 
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

ggplot(data = dt_houses, aes(x = factor(bathrooms), y = price)) +
  geom_boxplot() + 
  theme_minimal() 

here it is also almost obvious, that, if we have more bathrooms, price will be also up. Only one disadvantage, that in my dataset I do not have enough data about properties with 3 or 4 bathrooms, I have some on 3, but really luck on 4.

Furnishing is also important, many people search for apartments with furniture, but furniture could be not in a best shape or buyer may do not like the style. So from my opinion, it is not as strong(in prediction), as for example area.

How much real estate furnished or not:

ggplot(data = dt_houses, aes(x = factor(furnishingstatus), fill = factor(furnishingstatus))) + 
  geom_bar(color="#ced4da", alpha=0.25) + 
  scale_fill_viridis_d(option = "D") + 
  labs(title = "Bar Chart with Different Colors", 
       x = "Furnishing Status", 
       y = "Count") + 
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

We can see, that most of the houses are semi-furnished. which is also logical, because when we sell a house or apartment, probably we would take in most of the cases the most valuable things for us and furniture included.

Now, it would be great, to look at price and area distribution in differently furnished properties

ggplot(data = dt_houses, aes(y = price, x = area)) + 
  geom_point(data = dt_houses, aes(y = price, x = area, color = bedrooms)) +
  geom_hline(yintercept = mean(dt_houses$price), linetype='dashed') + 
  facet_grid(.~furnishingstatus) +
  scale_y_continuous(labels = label_number(scale = 1e-6, suffix = "M")) +
  scale_color_distiller(type = "seq", palette = "Greens") +
  theme_minimal() + 
  theme(axis.line = element_line(color = "#000000"))

Also, on average, you can notice, that unfurnished houses, are less expensive.

We can also take a look on some pie charts:

dt_mainroad_counts <- as.data.frame(table(dt_houses$mainroad)) #table() - creates frequency table
colnames(dt_mainroad_counts) <- c("mainroad_status", "count")
dt_mainroad_counts$percentage <- round(dt_mainroad_counts$count / sum(dt_mainroad_counts$count) * 100, 1)

ggplot(data = dt_mainroad_counts, aes(x = "", y = count, fill = mainroad_status)) +
  geom_bar(stat = "identity", width = 1, color = "white") +
  coord_polar("y", start = 0) +
  geom_text(aes(label = paste0(percentage, "%")), 
            position = position_stack(vjust = 0.5), color = "white", size = 4) +  
  theme_void() +  
  scale_fill_manual(values = c("#F1B147", "#47B1F1")) + 
  labs(
    title = "Distribution of Mainroad Status",
    fill = "Mainroad Status"
  )

Almost 86 percent of houses have main road, so maybe this won’t be a strong predictor variable.

dt_airconditioning_counts <- as.data.frame(table(dt_houses$airconditioning)) #table() - creates frequency table
colnames(dt_airconditioning_counts) <- c("airconditioning_status", "count")
dt_airconditioning_counts$percentage <- round(dt_airconditioning_counts$count / sum(dt_airconditioning_counts$count) * 100, 1)

ggplot(data = dt_airconditioning_counts, aes(x = "", y = count, fill = airconditioning_status)) +
  geom_bar(stat = "identity", width = 1, color = "white") +
  coord_polar("y", start = 0) +
  geom_text(aes(label = paste0(percentage, "%")), 
            position = position_stack(vjust = 0.5), color = "white", size = 4) +  
  theme_void() +  
  scale_fill_manual(values = c("#F1B147", "#47B1F1")) + 
  labs(
    title = "Distribution of Airconditioning status",
    fill = "Airconditioning Status"
  )

Here 68.4 percent has airconditioning, but I do not know, how it will affect predictions.

I think that would be enough exploration and we can start with models.

Models 1 & 2

• Evaluate and compare your models based on a reasonable evaluation metric of your choice. You must use the same metric for both models. Report both the training and the CV loss.

First, I would like to start pretty simple with linear model.

I consider to take all variables to my model, because they all seem to be very important.

Linear model

I will use lm function in R to find needed beta coefficients and create my model

price_lm <- lm(formula = price ~ area + bedrooms + hotwaterheating + airconditioning + stories + mainroad + parking + furnishingstatus + bathrooms + guestroom + basement + prefarea, data = dt_houses)

summary(price_lm)

Call:
lm(formula = price ~ area + bedrooms + hotwaterheating + airconditioning + 
    stories + mainroad + parking + furnishingstatus + bathrooms + 
    guestroom + basement + prefarea, data = dt_houses)

Residuals:
     Min       1Q   Median       3Q      Max 
-2619718  -657322   -68409   507176  5166695 

Coefficients:
                                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)                      42771.69  264313.31   0.162 0.871508    
area                               244.14      24.29  10.052  < 2e-16 ***
bedrooms                        114787.56   72598.66   1.581 0.114445    
hotwaterheatingyes              855447.15  223152.69   3.833 0.000141 ***
airconditioningyes              864958.31  108354.51   7.983 8.91e-15 ***
stories                         450848.00   64168.93   7.026 6.55e-12 ***
mainroadyes                     421272.59  142224.13   2.962 0.003193 ** 
parking                         277107.10   58525.89   4.735 2.82e-06 ***
furnishingstatussemi-furnished  -46344.62  116574.09  -0.398 0.691118    
furnishingstatusunfurnished    -411234.39  126210.56  -3.258 0.001192 ** 
bathrooms                       987668.11  103361.98   9.555  < 2e-16 ***
guestroomyes                    300525.86  131710.22   2.282 0.022901 *  
basementyes                     350106.90  110284.06   3.175 0.001587 ** 
prefareayes                     651543.80  115682.34   5.632 2.89e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1068000 on 531 degrees of freedom
Multiple R-squared:  0.6818,    Adjusted R-squared:  0.674 
F-statistic: 87.52 on 13 and 531 DF,  p-value: < 2.2e-16

We got 0.68 R-squared, which is not that bad for a model just made up. But that’s not all, I will try to do better here, but first, another model.

But I would like to measure performance of my models with RMSE, so I will calculate RMSE for linear model.

price_lm_rmse <- mean(sqrt(abs(price_lm$residuals)))

price_lm_rmse

[1] 797.382

Tree Model

I think this model could perform better, because there some variables which can affect this model not only linearly, but the other way, in this case tree model can show better performance.

In this coursework will be used rpart to create a regression tree.

prices_tree <- rpart(data = dt_houses, formula = price ~ area + bedrooms + hotwaterheating + airconditioning + stories + mainroad + parking + furnishingstatus + bathrooms + guestroom + basement + prefarea, method = 'anova')

prp(prices_tree, digits = -3)

printcp(prices_tree)

Regression tree:
rpart(formula = price ~ area + bedrooms + hotwaterheating + airconditioning + 
    stories + mainroad + parking + furnishingstatus + bathrooms + 
    guestroom + basement + prefarea, data = dt_houses, method = "anova")

Variables actually used in tree construction:
[1] airconditioning  area             basement         bathrooms        furnishingstatus parking         

Root node error: 1.9032e+15/545 = 3.4921e+12

n= 545 

         CP nsplit rel error  xerror     xstd
1  0.304946      0   1.00000 1.00372 0.085248
2  0.094553      1   0.69505 0.73174 0.063521
3  0.053743      2   0.60050 0.61952 0.054756
4  0.026381      3   0.54676 0.60650 0.052185
5  0.024922      4   0.52038 0.62465 0.054037
6  0.022993      5   0.49546 0.63581 0.055852
7  0.021374      6   0.47246 0.61705 0.054637
8  0.015261      7   0.45109 0.58917 0.053114
9  0.013952      8   0.43583 0.58121 0.052928
10 0.012386      9   0.42188 0.57898 0.052584
11 0.010000     10   0.40949 0.55610 0.050742

Now after I have built with the help of rpart tree model based on my dataset, let us explore it:

prices_tree

n= 545 

node), split, n, deviance, yval
      * denotes terminal node

 1) root 545 1.903208e+15 4766729  
   2) area< 5954 361 6.066751e+14 4029993  
     4) bathrooms< 1.5 293 3.297298e+14 3773561  
       8) area< 4016 174 1.437122e+14 3431227  
        16) furnishingstatus=unfurnished 78 4.036605e+13 2977962 *
        17) furnishingstatus=furnished,semi-furnished 96 7.430067e+13 3799505 *
       9) area>=4016 119 1.358098e+14 4274118 *
     5) bathrooms>=1.5 68 1.746610e+14 5134912  
      10) airconditioning=no 44 7.024826e+13 4563682 *
      11) airconditioning=yes 24 6.373358e+13 6182167 *
   3) area>=5954 184 7.161564e+14 6212174  
     6) bathrooms< 1.5 108 2.869179e+14 5382579  
      12) airconditioning=no 65 1.170629e+14 4843569  
        24) basement=no 38 5.226335e+13 4304816 *
        25) basement=yes 27 3.824662e+13 5601815 *
      13) airconditioning=yes 43 1.224240e+14 6197360 *
     7) bathrooms>=1.5 76 2.492851e+14 7391072  
      14) parking< 1.5 51 7.184700e+13 6859794 *
      15) parking>=1.5 25 1.336772e+14 8474878  
        30) airconditioning=no 10 5.146311e+13 7285600 *
        31) airconditioning=yes 15 5.864106e+13 9267729 *

We can see, that we have 31 Nodes, I think for this kind of dataset it may be okay.

Now it would be great to prune the tree, because I do not want my tree to overfit:

plotcp(prices_tree)

This is complexity of this tree. We need the lowest complexity, to get as few leafs as possible to get the best performance, so that tree won’t overfit the data.

prices_tree_min_cp <- prices_tree$cptable[which.min(prices_tree$cptable[, "xerror"]), "CP"]
model_tree <- prune(prices_tree, cp = prices_tree_min_cp )
prp(prices_tree,digits = -3)

after we pruned the tree, let’s calculate the RMSE for the tree model

prices_tree_pred <- predict(prices_tree, dt_houses[, c("area","bathrooms", "bedrooms", "hotwaterheating", "airconditioning", "parking", "stories", "mainroad", "furnishingstatus", "guestroom", "basement", "prefarea")])
prices_tree_rmse <- mean(sqrt(abs(dt_houses$price - prices_tree_pred)))

prices_tree_rmse

Ensemble

• Repeat the analysis one ensemble method of your choice.
• Investigate the hyperparameter settings of your ensemble with regards to your evaluation metric.
• Report both the training and the CV loss.
• Select the best configurations of your ensemble model based on the same evaluation metric as before.

Neural Network

• Repeat the analysis with a neural network.
• Investigate three different configurations with regards to your evaluation metric and select the best configuration. Use the same evaluation metric as before.
• Report both the training and the CV loss.

Model Comparison

• Compare the performance of the 4 models.

PCA

• Run a PCA on your input variables and discuss the scope for dimensionality reduction in your dataset.
• Rerun the previous 4 models on all PCs or on a reduced number of PCs.

Model Selection

• Compare all of your models based on their CV performance. You will have 8 options to consider: four models with and without PCA.
• Present the results in a table or chart.
• Estimate the expected loss of your best model on the test set.

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